The opening 45 minutes was a stalemate, with neither goalkeeper called upon to make a save of any description.
It took 57 minutes before Chris Smith was forced to concede a corner to deny Michael McGowan and the opener arrived soon after.
Kirk Broadfoot fouled Tom Brighton and Smith once again had to concede a corner from Eddie Malone's well struck free-kick.
Stephen O'Donnell took the flag-kick and Craig Higgins headed across goal for Alex Williams to meet the ball with his head at the far post.
Saints brought on Charlie Adam for Simon Lappin and the substitute scored the equaliser within a minute of joining the play.
David van Zanten provided the throughball which John Sutton touched on to Adam, who appeared to use a hand before slotting past Paul Jarvie.
This was Saints' first shot on target and within 10 minutes the visitors were ahead.
Broadfoot supplied the initial pass, Marc Corcoran the cross and Sutton the tap-in from close range for the winner.
Clyde had enough clear chances to turn it around with Craig Bryson and Roddy Hunter mainly responsible for allowing Saints to escape with the points.